The only injuries I've ever done myself while playing have been a general reduction in sensitivity of fingertips from the repeated impact (and even that is questionable: I just seem to have better tolerance of hot cups of tea, say, than friends who don't play the piano, but it might be for other reasons) and the very, very occasional broken fingernail when I've carelessly caught a finger in front of a black key - painful but a once-every-five-years kind of event (that's once in five years for someone who regularly plays 5-8 hours a day).
For small cuts I use styptic pencil, for big ones a sticking plaster. But never in 39 years of piano playing have I drawn blood as a direct result of playing the piano!